# Shortcut for exponential shrinkage

Remember the easy method for calculating exponential growth?  In case you don’t, here it is again:

1. Find a number to multiply by the original balance by converting the percentage to decimal and adding 1 (i.e., 5% becomes 1.05)
2. Find the number of years needed, i.e., 10
3. Raise the multiplier to the power of the number of years needed, i.e., 1.0510
4. Multiply by the original balance, i.e., \$1000 × 1.0510

You might think there would be a similar procedure for exponential shrinkage, and you would be right.  The only real trick is #1 above.  How do you convert 5% loss into a decimal number? I have seen people try all kinds of strange numbers here. For example, I have seen students multiply \$1000 by -0.05, resulting in a year 2 balance of -\$50. I have also seen students multiply \$1000 by 0.05 and get a year 2 balance of \$50.

If you can’t quite remember the right formula, always check to see if your answer makes sense!!  If I start with \$1000 and spend 5%, I can’t possibly be left with only \$50 at the end of the year – or even worse, -\$50.

OK, end of lecture.  So here is how you do want to think about it: a 5% loss is the same as a -5% growth, right?  So, I will have 100% of my money left, MINUS 5%.  In other words

100% - 5% = 95% = 0.95.

Remember, rates of shrinking are the same as NEGATIVE growth rates, and use the same formula to find the multiplier. So, some practice with negative growth rates:

1. If my IRA loses 7% every year...

What will \$1000 be worth
2. Unlike the developing world, population is actually falling in many developed countries. In Eastern Europe, for example, "growth" rates are as low as -0.5%. If the population of Bulgaria was 7.5 million in 2002, then what would its predicted population be in 2020?

3. Over the last 400 years, there have been 89 documented mammalian extinctions, out of about 5000 mammal species. This works out to a rate of -0.0045% per year.
be careful -- the extinction rate is less than 1% per year, so the multiplier should be > 99% !
4. The price of hard-drive space for personal computers has fallen rapidly over the last 4 decades. In 1981, the average price per meg of hard drive space was \$350 (yes, that means your little 1 gig thumb drive should be worth \$350,000 -- of course, they had disco back then too...)
By the next year, the price had fallen to \$250 per meg. Assuming that the shrinking prices of hard drives are exponential, answer the following questions: